Twin-width IV: ordered graphs and matrices

TL;DR

The paper delivers a comprehensive framework for bounded twin-width in hereditary, totally ordered binary structures, establishing multiple equivalent characterizations that bridge combinatorics, logic, and complexity. It extends the Stanley-Wilf/Marcus-Tardos growth dichotomy to matrix classes over finite alphabets and derives a sharp speed gap for ordered graphs, including precise growth bounds and structure obstructions. It also provides a fixed-parameter approximation for twin-width in ordered graphs, proves tractability results for FO model checking under bounded twin-width, and offers model-theoretic characterizations (e.g., monadic NIP, restrained classes) that unify width, transductions, and definability. Collectively, the results illuminate the fundamental role of twin-width as a dividing line for tractability and for understanding the growth and definability properties of ordered structures.

Abstract

We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollobás, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width.

Figures (11)

• Figure 1: The four bipartite graphs $G^n_=$, $G^n_\leqslant$, $G^n_\geqslant$, and $G^n_{{\not=}}$, for $n=6$.
• Figure 2: The six ordered graphs of $\mathscr M_{{\rm{s}},0,0}$ for ${\rm{s}}\in\{{=, {\not=}, \leqslant_l, \geqslant_l,} {\leqslant_r, \geqslant_r}\}$ corresponding to the same ordered matching of $\mathscr M_{=,0,0}$ represented to the left. In each ordered graph the black edges are those implied by the bold edge $uv$ in the matching. To picture the other classes $\mathscr M_{{\rm{s}},\rho,\lambda}$ with $(\lambda,\rho) {\not=} (0,0)$, one just needs to turn the left part and/or the right part of each graph into cliques.
• Figure 3: The matrices in $\mathcal{F}_=, \mathcal{F}_{{\not=}}, \mathcal{F}_{\leqslant_R}, \mathcal{F}_{\geqslant_R}, \mathcal{F}_{\leqslant_C}, \mathcal{F}_{\geqslant_C}$ (from left to right) for the same permutation matrix (the one to the left). The 1 entries are represented in black, the 0 entries, in white. As is standard with permutation patterns, we always place the first row of the matrix at the bottom.
• Figure 4: A bird's eye view of the paper. In green (arrows without a reference to a part of the paper), the implications that were already known for general binary structures. In red (other arrows except \ref{['it:bd-tww']}$\Rightarrow$\ref{['it:lin-mc']}), the new implications for matrices on finite alphabets, or ordered binary structures. The effective implication \ref{['it:bd-tww']}$\Rightarrow$\ref{['it:bd-rd']} is useful for \ref{['thm:approx-tww']}. See \ref{['fig:zoom']} for a more detailed proof diagram, distinguishing what is done in the language of matrices and what is done in the language of ordered graphs.
• Figure 5: Left: The adjacency matrix of the ordered graph $G$ with vertices $1,\ldots,n$ and edges $ij$ such that $i+j$ is odd, along the usual order. (The first row is at the bottom.) Right: The adjacency matrix along another order, encoding the adjacency as well as the original order. Every $4$-division contains some constant zone.

Theorems & Definitions (82)

• Theorem 1
• Theorem 2
• Theorem 3
• Conjecture 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 8
• Corollary 9
• Lemma 10